From Wikipedia:
The name for this region, apparently after the Bronx River, first appeared in the Annexed District of the Bronx created in 1874 out of part of Westchester County and was continued in the Borough of the Bronx, which included a larger annexation from Westchester County in 1898. The use of the definite article is attributed to the style of referring to rivers.
Jim
Very interesting. Thinking about this, I'd have thought that this explanation could have applied to Harlem as well, given that there is a Harlem River. However, the derivation of the name Harlem is not originally from a river, but from a city, as follows:
"The first European settlement in what is now Harlem was by Dutch settlers and was formalized in 1658 as Nieuw Haarlem (or New Haarlem), after the Dutch city of Haarlem."
The river's name came after the city's name, not the other way around, as with the Bronx. If Harlem's name derivation had instead been from the river, it could have been the District of the Harlem, then the Borough of the Harlem, and then the Harlem.
Also historically important in the use of "the" with Bronx is that Bronx originally referred to a region - like the Highlands, or the Everglades.
Enough wandering thoughts about that. Here's the RR bridge tie-in: 
The Broadway Bridge, connecting Manhattan with Da Bronx, spanning the Harlem River.-
